Ifthepickedlabeliscorrect, theagentgetsarewardofr = 0,andtheepisode ends, and ifthe picked label isincorrect, then the agent gets areward ofr = 1,and the episode continues to the next time-step (where it must guess another label for thesameimage). For the variant labelled "Adaptive", we train a classifierpθ(y|x)on the training dataset of images with the same architecture as the DQN agent. Clearly,thepolicy"alwaysswitch" is optimal inMA and so is -optimal under the distribution on MDPs. The proof is a simple modification of the construction in Proposition 5.1. Effectively, this policy either visits the left-most state or the rightmost state inthe final level.

The matrix-based Renyi's {\alpha}-order entropy functional was recently introduced using the normalized eigenspectrum of an Hermitian matrix of the projected data in the reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Renyi's {\alpha}-order entropy functional only defines the entropy of a single variable or mutual information between two random variables. In information theory and machine learning communities, one is also frequently interested in multivariate information quantities, such as the multivariate joint entropy and different interactive quantities among multiple variables. In this paper, we first define the matrix-based Renyi's {\alpha}-order joint entropy among multiple variables. We then show how this definition can ease the estimation of various information quantities that measure the interactions among multiple variables, such as interactive information and total correlation. We finally present an application to feature selection to show how our definition provides a simple yet powerful way to estimate a widely-acknowledged intractable quantity from data. A real example on hyperspectral image (HSI) band selection is also provided.